On the projective classification of smooth n-folds with n even

(1)

On the projective classification of smooth n-folds with n even

M. L. Fania and A. J. Sommese

Let Y ~ P c be an irreducible, n dimensional, projective variety with a smooth normalization a: M-*Y and let ~ = a * O p ( 1 ) r . Recent results o f [5], [6], [16]

imply that either (M, ~ ) is one of a list o f specific, well understood polarized varieties or there is a projective manifold X and an ample line bundle L on X such that:

a) M is the blowup n: M--,-X of X at a finite set F,

b) KM|174 "-1) where Kx@L "-1 is ample and spanned by global sections,

c) L = [ n ( S ) ] for a smooth s~l~el, or equivalently ~=rc*(L)@[rc-l(F)] -1, d) Kx@L "-2 is semi-ample and big, i.e. some positive power (Kx@Ln-~) t is spanned by global sections and the map ~: X-~Pc associated to F((Kx| has an n dimensional image.

The pair (X, L) is called the first reduction o f (M, ~e) and is very well be- haved, see [12], [14] and [17]. It is easy to convert information between (X, L) and (M, ~e).

Let ~ o s = ~ be the Remmert--Stein factorization o f the map a (in d) above) where r X--,-X" has connected fibres for a normal projective X', and s: X ' ~ P c is finite to one. There is an ample line bundle ~ on X ' such that r174 "-2.

The pair (X', ,~f') is known as the 2 nd reduction and the map 9 is called the second adjunction map. Such pairs have been studied by the authors [4], [15]. X" has only isolated singularities, is 2-factorial and Gorenstein in even dimensions. Thus for n even, n->4, .r174 ''-~ where for a smooth AEIL[, 2~(A) is Cartier, i.e.

[2~(A)] is invertible and L ' is 2-Cartier. This pleasant circumstance makes the 2 "a reduction almost as easy to use as the first reduction when n is even, and allows us to use the results o f Fujita [5] in this case. Combining this with a recent result [2] we can push the known classification a good deal further. To state our main result it is useful to recall the notion o f the spectral value o f a pair.

(2)

Let ~f be a nefand big line bundle on a normal projective variety ~ of dimension n=>l. In [16] the spectral value, a(~ r, of), of the pair (~r, of) is defined as the smallest real number z such that given any fraction p/q>~,/'(Kff|

for all integers N>-0 with qlN.

Note that a(M, La)=cr(X, L ) = a ( X ' , L')>-O.

The normalization used in the above definition of spectral value is very useful in organizing the known results, e.g. a(M, .oq')=0 if and only if (M, .~q~)=(P", OF,(1)), and a(M, L~~ if and only if (M, N')=(Q, Oo(1)) where Q=p.+X is a quadric or (M, ~ ) is a scroll over a curve. The known classification is for e(M, ~ ) ~ 3 . In [2] this is pushed to a(M, ~ ) < 4 - 3 / ( n + 1 ) . Our main result is

1.1. Theorem. Assume a(M, ~ ) > 3 (see [16]for the case a(M, Se)<=3). Let M be of dimension n where n is even and n~=4. Either h~174162

in which case a(M, s l) and K x , | ~-1 is semi-ample and big or (X', I,') is one of the following list:

a(X', L') = 3-~ and (X', L') 1 = (p4, Or,(3)).

a(X', L') = 3-ff and either a), b), e), or d), holds: 1

a) (X', L') = (pe, OF. (2)),

b) Kx,4=L "a, dim X ' = 4 , and there is an ample line bundle H on X" with

H a = Kx ,x,

c) there exists a holomorphic map ~: X ' ~ C , where C is a curve, K~,| "z=

~ * E for an ample line bundle E on C. Further the general fibre of ~ is (Q~, OQ~ (2)),

d) there exists a holomorphic map ~: X ' ~ S , where S is a surface, K~x,|

L ' Z : ~ * E for an ample line bundle E on S. Further the general fibre of ~P is

op,(2)).

3 r p l 0

a ( X ' , L ' ) = 3 ~ and K x, = ~ , d i m X ' = 6 , and there is an ample line bundle H on X" with H S = K x 1.

By passing to a general HE 1~1 we get information about odd dimensional M.

1.1.2. Corollary. Assume that a(M, s176 Let M be of dimension n where n is odd and n >= 5. Either

[ ( ( n - 1)~+ 1)K~+ {n3--5n2+ 9n--4)s 9 and

[ ( ( n - 1) ~ + 1)Kx+(n 8 - 5n~ + 9n--4)L] 9 L are effective, or one of the following is true:

a) (X', L') is the cone on (p6, Ope(2)), b) X" i~ 5 dimensional and K~,| '~~ Ox,,

(3)

On the projective classification of smooth n-folds with n even 247 c) there is a morphism ~Y: X-~C where C is a curve, L F = O p , ( 2 ) for a gen-

eral fibre F which is biholomorphic to P~, d) X" is 7 dimensional and KS,|

To illustrate the use o f these results which actually requires only that s be ample and spanned on M we give a single representative application in w 2. Let M be an n-fold with n-->4 and assume that there is a family o f lines on M w i t h a line through most and hence all points of M. Let t+n--1 be the dimension of the family where t ~ 0 by the hypothesis on the last line. Then (M, s has a 2 nd reduction on the above lists if

n ( n - 1 ) ( n - - 2 ) < ( t + 2 ) ( n ~ + l ) and n is even, or if

n 3 - 5 n ~ + 9 n - 4 < ( t + 2 ) [ ( n - 1)2+ 1] and n is odd.

This should be contrasted with the work o f Sato [10].

If .L~ ~ is very ample and the variety o f singular hyperplane sections

~eclL~l

has codimension k + 1 then using a theorem of Ein's ([3], see (0.5) for a statement and short proof) it follows that (M, ~ ) has a 2 nd reduction on the list for k > 0 if

n ( n - 1 ) ( n - 2 ) < ( . n + k + 2 - } ( n 2 § n is even

n3 - 5nZ + 9 n - 4 < ( n + k + 2

) ( ( n - 1 ) ~ + l ) ; n is odd.

Thus we are reduced to studying varieties on the above list and that o f [16]

if n is even and k ~ n - 7 (see also [8]).

It should be noted that the detailed classification of varieties on the lists with a special property, e.g. defect k discriminant locus, requires some further work.

We would like to thank the M. P. I. and the Italian Government for making this research possible. The second author's research was partially supported by the University o f Notre Dame, the National Science Foundation (DMS 84 30315 and DMS 87--22330), and the Max Planck Institut fiir Mathematik in Bonn.

O. Background material

Throughout this paper (M, ~ ) will denote a pair, consisting o f a smooth projective n-fold M, and an ampIe and spanned line bundle .oqe on M such that the map X-~Pc given by F(s is generically one to one.

0.1. Let L be a line bundle on M. We say that L is n e f i f cl(L).[C]>=O, for all effective curves C on M. We say that a nef line bundle L is big if cl(L)n>0. We

(4)

say that L is semi-ample if there exists an m > 0 such that

BslmL[,

the base locus of

ImLI,

is empty.

0.2. A reduction (X, L) of a pair (M, .oge) is a pair (X, L) consisting of an ample line bundle L on a projective manifold X such that:

a) M is the blowup 7t: M-~X of X at a finite set F,

b) La=~*(L)| -1 or equivalently

KMQ~"-I=rc*(KxQL'-I).

The pair (X, L) is also called the 1st reduction of (M, Se) if

K x Q L "-1

is ample.

For the following theorems we refer to [5], [6], [12], [13], [16].

Theorem 0.3.

Let (M, .L~') be as above. Then there exists a reduction

(X,

L) o f (M, s such that K x Q L "-~ is ample and spanned by global sections unless one o f the following holds:

a) (M, ~e)=(e", 0p41)) or (P', Opt(2)),

b) (M, s OQ.(1)),

where Q" is a smooth hyperquadric in

P"+~, c)

(M, ~ ) is a scroll over a smooth curve,

d) (M, 5e)

is a d e l Pezzo manifold, i.e. K ~ | "-1 =ON, e) (M, ~ ) is a hyperquadric fibration over a smooth curve,

f)

(M, ~ ) is a scroll over a surface.

Theorem 0.4.

Let (M, .oq') be as above. Assume that

dim

M=n~=3. I f (M, ~ce) is not as listed in

0.3

there exists a reduction (X, L) of (M, Le) such that K x | "-2 is semi-ample and big unless one of the following holds:

a) (X, L ) = ( P ~, O~,,(2)) ^{o r} (V 3, O~,3(3)),

b) (iV, L ) = ( Q z, Oo~(2)),

c)

there is a holomorphic surjection q~ : X ' ~ C onto C, a smooth curve where K~,QL3~.~o*~ for an ample line bundle ~ on C; in particular the general fibre of ~ is

(p2, Oa,,(2)) '

d)

K x ~ L -("-2), i.e. (X, L) is a Fano manifoM of co-index

3 (see [9]),

e) (X, L) is a d e l Pezzo fibration over a curve,

f)

(X, L) is a hyperquadric fibration over a surface,

g) n=>4

and (X, L) is a scroll over a threefold.

We need the following basic result (see [4], [15]).

Theorem 0.4.1.

Let (M, .~) be as above. Assume there is a reduction (X, L) with K x | n-~ ample. Assume that K x | "-~ is semiample and big and that n>=4.

Let ~: X ~ X " be the second adjunction map, i.e. there is a b#'ational morphism, 9 , and an ample line bundle ~Y" on X', a normal projective variety, such that #)* JY" = K x | "-2. Then X" has isolated singularities. Precisely there exists an algebraic set Z c X ' such that

dimZ=<l

and

~x-,-~(z):

X - - ~ - ~ ( Z ) ~ X ' - Z is a biholo- morphism. I f C c Z is a pure one dimensional subvariety, then C is smooth, CcX;~g,

(5)

On the projective classification of smooth n-folds with n even ~ 9 and in a neighborhood U of C, ~: ~ - I ( U ) - ~ U is simply the blowup of C. I f x is a zero dimensional irreducible component of Z then 4)-X(x) is one of the following,

a) p , - 1 with normal bundle Op,_~(-2), Le,_x=Ol,,_~(1),

b) Q biholomorphic to an irreducible quadrie in P~ with LQ=Op,(1)O , and normal bundle

Opn(-- 1)Q.

Letting Z = Z ~ + Z 2 where Z l = s e t of points x with ~-a(x) as in a) and Z 2 = Z - Z 1 , and letting L ' = ( ~ , L ) * * it can be seen that using Q-Cartier divisors 9 * L / - t~-1 ( Z 2 ) - 1/2~-1(Z1) = L .

0.4.2. Let L'= (q),L)**. This is a 2-Cartier divisor. Indeed except for points x with ~ - l ( x ) of the form a) it is Cartier. Similarly Kx, is 2-Cartier and Cartier if n = d i m X ' is even. We often write L '2a for the line bundle (2L')L

0.4.3. Often we will have a surjective map f : X ' ~ Y where ~ 0 < d i m Y <

dim X'. I f dim Y > I then by 0.4.1 we choose a general fibre F o X " o f f such that ~ gives a biholomorphism of ~b-~(F) and F. Thus F can be identified with a general fibre o f f o ~ : X ~ Y.

I f dim Y = I , then a general fibre F o f f is smooth, FcX~,g and meets the set Z of 0.4.1 in a finite number of points 5 ~ {x~ . . . . , x , } c F obtained by intersecting F with a smooth curve C c Z . Note ~ : ~-~(F)--,F expresses ~ - ~ ( F ) as F with 6 ~ blown up and since Kx, F=KF,

n - 2

K~-a<F)| = ~*(KF|

Lemma 0.4.4. Let ~Y" be a nef and big line bundle on a normal projective Goren- stein variety Y. Assume Irr (Y) is finite and ( K ~ | for some a > 0 , b > 0 , t > 0 . Then K;| Further b / a ~ n + l .

Proof. Choose the smallest integer t > 0 such that (K~| Let q: Y ' ~ Y be the unramified cover associated to the t-th root of the constant func- tion. By choice of t, Y' is irreducible and K~,| where ~'=q*o~('.

Since K ; / = J g "-bt, K~,,=oU "-b we see that K~ -1, K~, 1 are nef and big. Thus by the Kawamata--Viehweg vanishing theorem (see [16], (0.2.1)), hi(Or)=hr

i > 0 . Thus 2(Or)=Z(Or)= 1. But since q is an unramified cover z(Oy,)=tx(Oy).

This implies t = 1.

To see that b/a<=n+ 1 is a simple modification o f an old Hirzebruch--Kodaira argument. Note that since c ( is nef and big, the polynomial p(t)=x(KrNoY "t) is an nth degree polynomial with nth degree term nonvanishing. By the Kawa- mata--Viehweg vanishing theorem used above, p(t)=h~174 t) for all t > 0 . I f b / a > n + l , then (Kr|174 has a n e f a n d big inverse for t between 1 and n + 1. Thus we have the absurdity t h a t p ( t ) = h ~ t) = 0 for n + 1 integer values. []

(6)

0.5. Let s be a very ample line bundle on M. Let A c [Aal be the variety of singular hyperplane sections. I f A has codimension k + 1 then for a general point xEA, the set o f singular points o f the hyperplane section A corresponding to x is a linear go=pk, o f non degenerate quadratic singularities. Thus the two jet 9 o f a section sEF(.oq ~) gives on go a section o f Ngo*(2)| a which is non degenerate as a symmetric form at all points o f go. Thus

Theorem 0.5.1. (Ein.) N~| In particular given a line 2cgo, N z ~ Na/g a @Ngo,~ ~-Or,(1)+(k-1) @ Or, (1) +("-~)/~ @ Off~ <"-k)/2,

Thus

degK~,~ = - ( n + k + 2 ) ] 2 and 0 = (n+k) m o d 2 .

Remark. The parity result had earlier been observed by A. Landman. The number k is also called the defect o f M, def (M) (see [3] for details).

1. The main Theorem

Let s be an ample and spanned line bundle on a smooth projective manifold M.

Assume the map M-,-P c associated to F(~e) is generically one to one. Assume dim X=n is ->4 and even. Assume a(M, s176 and let X, L be as in 0.4. This section is devoted to proving the main theorem stated in the introduction.

Proof of Theorem 1.1. Recent results of [5], [6], [16] imply that the pair (M, s has a 2 na reduction (X', A r) unless (M, A o) is as listed in 0.3 and 0.4. It is easy to see from [4], [15] that X" has only isolated rational singularities and in fact X"

is 2-factorial and Gorenstein in even dimensions. Thus for n even, n->4, the ample line bundle ~f" is Kx,| ""-2, where L ' is as in the introduction.

Hence we can apply the results of Fujita [5], to the pair (X', J~f'). F r o m ([5], Thm. 1, 2) we see that Kx,| r"-I is nef unless

a) (X', g : ) = ( P " , O r , 0 ) ) ,

b) X ' is a hyperquadric Q" in p , + l and g{'=Oo,(1 ), c) (X', g : ) is a scroll over a smooth curve.

Noting that J : = K x , | "-2, in a) we have Kp.| Hence --(n+ l)+(n--2) d= l, where d E Z and d i s such that L'=Oe,(d). Using the ampleness of U , dis seen to be an integer > 0 . It follows that 3~n_<-6. By assumption n is even, thus we have either

aO (X', L') = (P~, 0~,(3)), or

as) (x', L') = (P", Op,(2)).

(7)

On the projective classification of smooth n-folds with n even 251 Note that L ' is Cartier in case b) since the only singularity o f 0.4.1 for which L ' would not be Cartier doesn't occur on hyperquadrics.

Identical reasoning can be carried out for b) and c) and we obtain in b) either

O r

b,) (X', L') = (QZ, Oos(4)),

b~) (X', L') = (Q~, OQs(2))

and in c) we see that n = d i m X ' = 3 , 5. Note that both b) and c) cannot occur since n is even. Thus

Kx,| ""-1

is nefunless (At', L 9 is as in at) and az). We will denote, for simplicity,

Kx,| ""-~

by d/. F r o m (2.6) o f [7] it follows that the linear system

Im~gl

is base-point free for all m>>0. Let 71 be the morphism associated to ImJg] for m>>0.

Let W = 7 / ( X ' ) . Note that d i m W < - 2 or

d i m W = n

(see [16]). To see that dim W<_-2 note that the restriction o f

,.r QJg "-~

to a generic fibre, F, o f 7 / i s Kr| -x and

(KF| "

if Oa, for some positive m. N o w use lemma 0.4.4.

d) I f dim W = 0 then

(Kx,| ,.

Thus by 0.4.4

Kx,| ,.

Thus

K~,|

I f n is relatively prime to ( n - 1)(n--2)/2 then there exists an ample H such that

H("-x)("-~)/~Kx, X=.

Since

(n-1)(n-1)/2~_

n + l and n=~4 and even we conclude that n = 4 . In this case

K4x,|

It is easy to see that if n and ( n - 1) ( n - 2)/2 have a common factor it is 2 and then

n/2, (n-1)(n-2)/4

are relatively prime. Thus there exists an ample line bundle H such that

H("-I)("-~)14=Kx3.

Since n=>4 and even and

(n--1)(n-2)/4<=n+l

by Lemma 0.4.4 we conclude that n = 6 . In this ease we have

HS=Kx, ~,

and H s = 2L'.

e) I f dim W = 1 and if we let F be a general fibre o f 7/we have (Kr | ~ - l ) t = Or for some t > 0 . By (0.4.4,

KF|

Since F is smooth as noted in 0,4.3, F is a smooth quadric Q c P " and Kv=OQ(1 ). Since

L'F=OQ(d)

for d a positive integer, this gives

- ( n - 1 ) + ( n - 2 ) d = l

or ( n - 2 ) ( d - 1 ) = 2 . Since n is =>4 and even, we conclude that n = 4 , d = 2 .

f) I f dim W = 2 and F denotes a general fibre o f 7 / then by 0.4.3 W is smooth.

We have

KF|

i.e. (F, ~ ) = ( p , - 2 , Op,-~(1)). As before we can see that n = 4 and d = 2 .

It is easy to see that

a(X',

L ' ) = 3 ~ if (X', L') is as in at), and

a(X',

L ' ) = 3 - s 1

if (X', L') is as in az), e), or f). In the first example o f d) ~r(X', L ' ) = 3 ~ ; in the second 3 ~ . Hence (X', L') is as in Theorem 1.1 above. 2

I f dim

W=n

then

.,'/r174 ~"-~

is nef and big. Consider the line bundle

Kx,|162

Either

hO(Kx,|

or

h~174162

(8)

In ([2], Theorem 2.2) it is shown that if aY'=/Cx| "-2 is nef and big and

h~174 ")

= 0 then there is a birational morphism ~: X-~P" with J~ff= ~* Or,(1 ).

The argument used there works for any line bundle J{" on a normal Y such that:

a) 2U t is spanned by global sections for all sufficiently large t, b) S is big,

c) h~(sC~s)=O for i > 0 , j > 0 .

Since X ' is Gorenstein with rational singularities Kawamata's base point free theorem and the fact that J l and J/( are nef and big imply a) and b). Since d C J = K x , @ ( ~ " - ~ | s-~) for j ~ l , dim S i n g ( X ' ) = 0 , and ~ , J// are nef and big, the Kawamata--Viehweg vanishing theorem implies c).

Thus if J g = ~* Or, U) , then r (L')**=

Oe,(d)

where - - ( n + 1 ) + ( n - - 1 ) ( ( n - 2 )

d - n -

1) : 1

or

( n - 1 ) ( ( n - 2 ) d - n - 2 ) = 3 .

Since n=>4, this implies n = 4 and 2d=7. This is clearly not possible.

Proof of Corollary 1.1.2.

Let

AC]LI

be a general element. Corollary 1.1.2 will follow from (1.1) if we show that

(A', L'A,)

can be one of the exceptions of (1.1) only if

a)

(X', L')

is the cone on (p6, O1,6(2)), b) X ' is 5 dimensional and

K~,@L'I~ Ox,,

c) there is a morphism ~g:

X~C

where C is a smooth curve, L r = O r , ( 2 ) for a general fibre F which is biholomorphic to p4,

d)

X"

is 7 dimensional and

K~,@L'26= Ox,.

Note if

A'=P 4,

then X ' is smooth in a neighborhood of

A'.

This follows by looking over the possible singularities of 0.4.1. Since

A"

is therefore Cartier and ample it follows from Scorza's theorem (see [1]) that X ' is a cone over P~. The only singularity on

X"

is the vertex. Checking the list in 0.4.1 it doesn't occur.

- - 4 t

I f

K~, =LA,~,

with d i m A ' = 4 ,

ACIL[,

A'=c~(A), then (K~|176 has a section zero only on the inverse image o f the positive dimensional fibre o f

A ~A"

and

hO((K~|

^Consider

O-+Kx|174174176174176

Since

KxNL ~

is nef and big by assumption, we conclude

hl(K~|

by the Kodaira vanishing theorem. Also since A is a general element of

ILl

and

h~174176

we conclude

h~174176

Thus

4Kx+IOL=D

where D

is an effective divisor supported on the set of positive dimensional fibre of ~ :

X~X'.

From this we conclude the Cartier divisor

4Kx,+

10L' is trivial.

Assume now that for

ACILI,

A ' = ~ ( A ) there exists a 7':

A'~C,

C a curve,

KJ,| 7SE*

for an ample line bundle E on C with general fibre F o f 7 j equal (Qa, OQ~(2)). By [11], the map 7~o~a:

A-~S

extends to a map f :

X--S.*

By 0.4.3 we can assume that for a general f i b r e f o f

X~-C, (f, LI)

has a first reduction

(9)

On the projective classification of smooth n-folds with n even 253 ( f ' , L),) with FC[L'f,[. Since K~| we conclude by the first Lefschetz theorem, (Ky,|174 Thus there is an ample line bundle H with H~---K-f, ~. Thus f , = p 4 . Since H2----L'r, L ) , = O v , ( 2 ).

Similarly the 4 th case leads to a map X ' - ~ S with KrZ| Ov for a general fibre F with dim F = 3 . This implies K ~ = H 5 for an ample line bundle H which is easily seen to be impossible. In the last case we conclude as in the 3 ~d case that 6 K x = 2 6 L = D where D is an effective divisor supported on the set o f positive dimensional fibre o f X-+X'. Thus K~,| Ox,. []

Theorem 1.2. Let Y c P c be an n dimensional irreducible projective variety whose normalization M is smooth of dimension n~=4. Assume that (M, H ) is not as listed in 0.3 and 0.4. Let S = (~1~_~<=,-~ Hi for the general HI6IH[. Then if n is even either

d e g M <- ( g - l ) 1-~ 2 n 2 _ n _ 1 and

n+3 )

Ks.L<- I + n~_--U_ 2) K~

or (M, H ) has a 2rid reduction (X', ~") such that (X', L') is as in Theorem 1.1.

I f n is odd then either

( ,+2 I

d e g M - < ( g - l ) 1~ 2n2_---~n+2. } and

K s . L < ( I + n + 2 ]

= t n~--3n) K~

or (M, H ) has a 2nd reduction (X', o~ff) such that (X', L') is as in Corollary 1.1.2.

Proof. F r o m 1.1 it follows that either h~174 or (M, ~ ) has a 2 nd reduction (X', ~ ) such that (X', L') is as listed in the T h e o r e m 1.1 or in 1.1.2.

If h~ then since (KM@H(n-2))s=K S and ~e is ample

we have

Ks: H >= ( n + l ) ( n - - 2 ) .oge.H.

n ~ + 1

By the adjunction formula and the above inequality we see that

2 n 2 - - n - 1 5 ~

2 g - - 2 = ( K s + ~ ) , ~ a = K s . ~ + S r => n ~ + l . ~ ,

(10)

Hence

n § ) d e g M = ~ e . ~ < = ( g - l ) 1+ 2 n 2 _ n _ l ) .

Similar reasoning with Corollary 1.1.2 yields the given result. []

Remark 1.2.1. Assume n_->4 and h~ Using Castelnuovo's bound for the genus of a curve in terms of its degree we get g_->8 and further

[ n+3 ^.)

a) if n is even then d e g M = < ( g - l ) 1-~ 2 n ~ _ n _ l ) ,

( n+2 /

b) if n is odd then degM<= ( g - l ) 1-~ 2 n 2 _ 5 n + 2 ).

2. An application

Proposition 2.1. L e t M be an n dimengional manifold. Assume that there is a family o f lines on M with at least a t >-0 dimensional subfamily o f lines through most points o f M , Then (M, 5~) is as in 0.3 or 0.4 or has a 2 nd reduction as in Theo- rem 1.1 or Corollary 1.1.2/f

n ( n - 1 ) ( n - 2 ) < (t+2)(nZ+ 1) and n is even ~ 4 n 3 - 5 n ~ + 9 n - 4 < ( t + 2 ) [ ( n - 1 ) 2 + l ] and n is odd ~_ 5.

Proof. Let 2 be a line through a general point p of M. Let Nx be the normal bundle of 2 in M. By hypothesis, N~ is generically spanned by global sections. Hence (2.1.1) Nz = Oi= 1 Ok al

.-1

with a~ => 0.

Let Iv/x denote the ideal sheaf on 2 of germs of holomorphic functions vanishing at p. Since hl(Na| where the Hilbert scheme A of lines in X through p is smooth at the point to corresponding to 2. Hence there is a unique irreducible

component A0 of the Hilbert scheme containing to. Also dim Ao = h~174 Ip/~) = ~ = ~ a,.

For simplicity we denote this dimension by t.

Unless (M, Lz) is as in 0.3 or 0.4 or has a second reduction Of', JC) as in 1.1 or 1.1.2 it follows that

a) ( n 2 + l ) K ~ + n ( n - - 1 ) ( n - - 2 ) ~ is effective i f n is even and ~ 4 ,

b) [((n-- 1)3+ 1)KM+(n 3 - 5n~+9n--4)La]. s is effective if n is odd and ~5.

(11)

On the projective classification of smooth n-folds with n even 255

By the a d j u n c t i o n f o r m u l a KM- 2 = - 2 - d e g (det N a ) = - - 2 - t. Since .c#. ,~= 1 it follows f r o m a) t h a t

~) - ( n 2 + l ) ( 2 + t ) + n ( n - 1 ) ( n - 2 ) >-0 if n is even a n d - > 4 , a n d f r o m b) t h a t

fl) - [ ( n - 1 ) ~ + l ] ( 2 + t ) + ( n S - 5 n Z + 9 n - 4 ) > - O i f n i s o d d a n d _->5. []

Proposition 2.2. L e t ~ be a very ample line bundle on an n-foM M with n->4.

Assume that d e f ( M ) = k > 0 . Then (M, .o~q') has a 2 no reduction as in Theorem 1.1 or Corollary 1 . 1 . 2 / f

n is even and n ( n - 1 ) ( n - 2 ) < ( n + k + 2 ) [ n 2 + l ] / 2 , or

n is odd and n a - 5 n 2 + 9 n - 4 < ( n + k + 2 ) [ ( n - 1 ) ~ + l ] / 2 .

Proof. F r o m 0.5.1 a n d the a d j u n c t i o n f o r m u l a it follows t h a t deg Kx, ' 4=

- - ( n + ~c + 2)/2.

H e n c e as in the p r o o f o f 2.1 we c o n c l u d e t h a t (M, Lr has a 2 "a r e d u c t i o n as in T h e o r e m 1.1 o r C o r o l l a r y 1.1.2 unless the a b o v e inequalities occur. []

Conjecture 2.3. L e t L be a very ample line bundle on a smooth connected projec- tire n-fold, X. Assume that the spectral value, a(X, L), o f the pair (X, L ) is <-n.

Then the only possible values o f a(X, L) are n + 1 - - P where p, q are integers sat- q

isfying O<q<-p<=n+l.

References

1. BADnSCU, L., On ample divisors, Nagoya Math. J. 86 (1982), 155--171.

2. B~LTRA~EX'rI, M., BIANCO~OR~, A. and SOMMES~, A. J., Projective n-folds of log general type, I, to appear Trans. Amer. Math. Soc.

3. ExN, L., Varieties with small dual varieties, I, lnv. Math. 86 (1986), 63---74; II, Duke Math. I.

52 (1985), 895--902.

4. FAr, A, M. L., Configurations of --2 rational curves on sectional surfaces of n-folds, Math.

Ann. 275 (1986), 317--325.

5. FUJXTA, T., On polarized manifolds whose adjoint bundles are not semipositive, Algebraic Geometry, Sendai, 1985. Advanced Studies in Pure Mathematics 10 (1987), 167--178.

6. ION~SCU, P., Generalized adjunction and applications, Math. Proc. Cambridge Philos. Soc. 9 (1986), 457--472.

7. KAWAMATA, Y., On the cone of curves of algebraic varieties, Ann. of Math. 119 (1984), 603-- 633.

8. LANTERI, A. and PALLESCI-II, M., Projective manifolds containing many rational curves, Indiana Univ. Math. Jour. 36 (1987), 857--865.

(12)

9. MUKAI, S., On Fano manifolds of co-index 3, preprint.

10. SATO, E., Varieties which contain many lines, Max Planck Institut preprint, 86--2.

11. SOM~n~SE, A. J., On manifolds that cannot be ample divisors, Math. Ann., 221 (1976), 55--72.

12. SOMM~SE, A. J., Hyperplane sections of projective surfaces 1. The adjunction mapping, Duke Math. Y. 46 (1979), 377--401.

13. SOMMESE, A. J., On hyperplane sections, Algebraic Geometry, Proceedings, Chicago Circle Conference, 1980, Springer Lecture Notes 862 (1982), 232 271.

14. SOMM~S~, A. J., On the minimality of hyperplane sections of projective threefolds, & reine und angew. Math. 329 (1981), 16--41.

15. SOMMESE, A. J., On the configurations of - 2 rational curves on hyperplane sections of projective threefolds. In: Classifications of algebraic and analytic manifolds. Progress in Math. 39 (1980), 465--497.

16. SOMMnSE, A. J., On the adjunction theoretic structure of projective varieties. Proc. of Com- plex Analysis and Algebraic Geometry Conference, ed. by H. Grauert, G~Sttingen,

1985 Springer Lecture Notes 1194 (1986), 175--213.

17. SOM~ES~, A. J. and VAN DE VEY, A., On the adjunction mapping, Math. Ann. 278 (1987), 593--603.

Received M a y 31, 1987 M. L. Fania

Dipartimento di Matematica Universita degli Studi dell'Aquila 67 100 L'Aquila

Italy

A. J. Sommese

Department of Mathematics University of Notre Dame Notre Dame, IN 46556 USA

bitnet: SOMMESE @ IRISHMVS